Formal, explicit ideas about arguments, inference and logic originated, within the Western tradition, with Aristotle. Aristotle wrote about these topics in several different works (including one called the Topics ;-) ). However, the most basic single principle is The Law of Non-contradiction, which can be found in various places, including Metaphysics book IV, chapters 3 & 4. A typical formulation is: " ...it is impossible for anything at the same time to be and not to be [in the same sense]" (1006 a 1). Its importance is stated slightly earlier, " ...this is naturally the starting-point even for all the other axioms" (1005 b 30). Forgive me for waxing philosophical, but this question by its nature has philosophical content that cannot simply be pushed aside for convenience.
Consider this thought-experiment: Alex flips a coin, catches it and turns it over onto his forearm with his hand covering the side facing up. Bob was standing in just the right position; he briefly saw the coin in Alex's hand, and thus can deduce which side is facing up now. However, Carlos did not see the coin--he wasn't in the right spot. At this point, Alex asks them what the probability is that the coin shows heads. Carlos suggests that the probability is .5, as that is the long-run frequency of heads. Bob disagrees, he confidently asserts that the probability is nothing else but exactly 0.
Now, who is right? It is possible, of course, that Bob mis-saw and is incorrect (let us assume that he did not mis-see). Nonetheless, you cannot hold that both are right and hold to the law of non-contradiction. (I suppose that if you don't believe in the law of non-contradiction, you could think they're both right, or some other such formulation.) Now imagine a similar case, but without Bob present, could Carlos' suggestion be more right (eh?) without Bob around, since no one saw the coin? The application of the law of non-contradiction is not quite as clear in this case, but I think it is obvious that the parts of the situation that seem to be important are held constant from the former to the latter. There have been many attempts to define probability, and in the future there may still yet be many more, but a definition of probability as a function of who happens to be standing around and where they happen to be positioned has little appeal. At any rate (guessing by your use of the phrase "confidence interval"), we are working within the Frequentist approach, and therein whether anyone knows the true state of the coin is irrelevant. It is not a random variable--it is a realized value and either it shows heads, or it shows tails.
As @John notes, the state of a coin may not at first seem similar to the question of whether a confidence interval covers the true mean. However, instead of a coin, we can understand this abstractly as a realized value drawn from a Bernoulli distribution with parameter $p$. In the coin situation, $p=.5$, whereas for a 95% CI, $p=.95$. What's important to realize in making the connection is that the important part of the metaphor isn't the $p$ that governs the situation, but rather that the flipped coin or the calculated CI is a realized value, not a random variable.
It is important for me to note at this point that all of this is the case within a Frequentist conception of probability. The Bayesian perspective does not violate the law of non-contradiction, it simply starts from different metaphysical assumptions about the nature of reality (more specifically about probability). Others on CV are much better versed in the Bayesian perspective than I am, and perhaps they may explain why the assumptions behind your question do not apply within the Bayesian approach, and that in fact, there may well be a 95% probability of the mean lying within a 95% credible interval, under certain conditions including (among others) that the prior used was accurate (see the comment by @DikranMarsupial below). However, I think all would agree, that once you state you are working within the Frequentist approach, it cannot be the case that the probability of the true mean lying within any particular 95% CI is .95.